Dynamic Analysis of a Multiple-Span Curved Truss Supported by Geometrical Nonlinear System

Abstract

The motivation of this paper is to improve the accuracy of modal test for large-scale curved space structure, which we propose a kind of quasi-zero-stiffness (QZS) supporting method and develop a multiple-span circle curved beam model supported by geometrical nonlinear system applied in the complicated dynamical analysis. The QZS system is applied in the dynamic analysis for the circle antenna seemed as a multiple-span circle curved beam, for which the equivalent material parameters and the governing equation are obtained by energy equivalent method and Hamilton’s principle, respectively. Meanwhile, the theoretical modal analysis is carried out by applying transformation matrix method from order 1 to 4 with different supporting parameters, showing that the results obtained from QZS supports are more consistent with the free-constraint condition than linear supports with a weaker influence of supporting stiffness on the modal results. Moreover, the forced vibration and complicated dynamic behaviors are also taken into consideration by the averaging method and the fourth-order Runge–Kutta method, which indicated the system will vibrate in a chaotic state through the process double-periodical and quasi-periodical state, and the first two modalities will enter in the vibrating state with the same periodicity, which has a wider vibrating range for modal 1 than modal 2. Finally, the dynamic model of the circle truss is verified by the finite element method (FEM) in Ansys 2021, explaining the correctness of the dynamic model. The results in this paper indicate more ideal characteristics of the proposed quasi-zero-stiffness supporting method of modal test for the large-scale circle truss than linear elastic methods, especially in low-order modalities, which can obtain consistent modal results between ground tests and its space missions, providing a reference for engineering application.

Appendix A

$$\begin{aligned} \left\{ \begin{aligned}&{{a}_{11}}=\cos \left( \alpha {{l}_{k}} \right) ,{{a}_{12}}=\sin \left( \alpha {{l}_{k}} \right) ,{{a}_{13}}={{e}^{{{\beta }_{1}}{{l}_{k}}}}\cos \left( \beta {{l}_{k}} \right) ,{{a}_{14}}={{e}^{{{\beta }_{1}}{{l}_{k}}}}\sin \left( \beta {{l}_{k}} \right) \\&{{a}_{15}}={{e}^{{{\gamma }_{1}}{{l}_{k}}}}\cos \left( \gamma {{l}_{k}} \right) ,{{a}_{16}}={{e}^{{{\gamma }_{1}}{{l}_{k}}}}\sin \left( \gamma {{l}_{k}} \right) , {{a}_{21}}=-\alpha \sin \left( \alpha {{l}_{k}} \right) ,{{a}_{22}}=\alpha \cos \left( \alpha {{l}_{k}} \right) \\&{{a}_{23}}={{e}^{{{\beta }_{1}}{{l}_{k}}}}\left( {{\beta }_{1}}\cos \left( \beta {{l}_{k}} \right) -\beta \sin \left( \beta {{l}_{k}} \right) \right) ,{{a}_{24}}={{e}^{{{\beta }_{1}}{{l}_{k}}}}\left( {{\beta }_{1}}\sin \left( \beta {{l}_{k}} \right) +\beta \cos \left( \beta {{l}_{k}} \right) \right) \\&{{a}_{25}}={{e}^{{{\gamma }_{1}}{{l}_{k}}}}\left( {{\gamma }_{1}}\cos \left( \gamma {{l}_{k}} \right) -\gamma \sin \left( \gamma {{l}_{k}} \right) \right) ,{{a}_{26}}={{e}^{{{\gamma }_{1}}{{l}_{k}}}}\left( {{\gamma }_{1}}\sin \left( \gamma {{l}_{k}} \right) +\gamma \cos \left( \gamma {{l}_{k}} \right) \right) \\&{{a}_{31}}=\frac{\cos \left( \alpha {{l}_{k}} \right) }{{{\eta }_{1}}},{{a}_{32}}=\frac{\sin \left( \alpha {{l}_{k}} \right) }{{{\eta }_{2}}}, {{a}_{33}}=\frac{{{e}^{{{\beta }_{1}}{{l}_{k}}}}\cos \left( \beta {{l}_{k}} \right) }{{{\eta }_{3}}},{{a}_{34}}=\frac{{{e}^{{{\beta }_{1}}{{l}_{k}}}}\sin \left( \beta {{l}_{k}} \right) }{{{\eta }_{4}}}\\&{{a}_{35}}=\frac{{{e}^{{{\gamma }_{1}}{{l}_{k}}}}\cos \left( \gamma {{l}_{k}} \right) }{{{\eta }_{5}}},{{a}_{36}}=\frac{{{e}^{{{\gamma }_{1}}{{l}_{k}}}}\sin \left( \gamma {{l}_{k}} \right) }{{{\eta }_{6}}},{{a}_{41}}=-\alpha \sin \left( \alpha {{l}_{k}} \right) \left( \frac{1}{R}+\frac{1}{{{\eta }_{1}}} \right) , \\&{{a}_{42}}=\alpha \cos \left( \alpha {{l}_{k}} \right) \left( \frac{1}{R}+\frac{1}{{{\eta }_{2}}} \right) ,{{a}_{43}}={{e}^{{{\beta }_{1}}{{l}_{k}}}}\left( {{\beta }_{1}}\cos \left( \beta {{l}_{k}} \right) -\beta \sin \left( \beta {{l}_{k}} \right) \right) \left( \frac{1}{{{\eta }_{3}}}+\frac{1}{R} \right) \\&{{a}_{44}}={{e}^{{{\beta }_{1}}{{l}_{k}}}}\left( {{\beta }_{1}}\sin \left( \beta {{l}_{k}} \right) +\beta \cos \left( \beta {{l}_{k}} \right) \right) \left( \frac{1}{R}+\frac{1}{{{\eta }_{4}}} \right) \\&{{a}_{45}}={{e}^{{{\gamma }_{1}}{{l}_{k}}}}\left( {{\gamma }_{1}}\cos \left( \gamma {{l}_{k}} \right) -\gamma \sin \left( \gamma {{l}_{k}} \right) \right) \left( \frac{1}{R}+\frac{1}{{{\eta }_{5}}} \right) \\&{{a}_{46}}={{e}^{{{\gamma }_{1}}{{l}_{k}}}}\left( {{\gamma }_{1}}\sin \left( \gamma {{l}_{k}} \right) +\beta \cos \left( \gamma {{l}_{k}} \right) \right) \left( \frac{1}{{{\eta }_{6}}}+\frac{1}{R} \right) \\&{{a}_{51}}=\cos \left( \alpha {{l}_{k}} \right) \left( \frac{1}{{{\eta }_{1}}R}+{{\alpha }^{2}} \right) ,{{a}_{52}}=\sin \left( \alpha {{l}_{k}} \right) \left( \frac{1}{{{\eta }_{2}}R}+{{\alpha }^{2}} \right) \\&{{a}_{53}}={{e}^{{{\beta }_{1}}{{l}_{k}}}}\left( \frac{\cos \left( \beta {{l}_{k}} \right) }{{{\eta }_{3}}R}-\left( {{\beta }_{1}}^{2}\cos \left( \beta {{l}_{k}} \right) -2\beta {{\beta }_{1}}\sin \left( \beta {{l}_{k}} \right) -{{\beta }^{2}}\cos \left( \beta {{l}_{k}} \right) \right) \right) \\&{{a}_{54}}={{e}^{{{\beta }_{1}}{{l}_{k}}}}\left( \frac{\sin \left( \beta {{l}_{k}} \right) }{{{\eta }_{4}}R}-\left( {{\beta }_{1}}^{2}\sin \left( \beta {{l}_{k}} \right) +2\beta {{\beta }_{1}}\cos \left( \beta {{l}_{k}} \right) -{{\beta }^{2}}\sin \left( \beta {{l}_{k}} \right) \right) \right) \\&{{a}_{55}}={{e}^{{{\gamma }_{1}}\theta }}\left( \frac{\cos \left( \gamma {{l}_{k}} \right) }{{{\eta }_{5}}R}-\left( {{\gamma }_{1}}^{2}\cos \left( \gamma {{l}_{k}} \right) -2\gamma {{\gamma }_{1}}\sin \left( \gamma {{l}_{k}} \right) -{{\gamma }^{2}}\cos \left( \gamma {{l}_{k}} \right) \right) \right) \\&{{a}_{56}}={{e}^{{{\gamma }_{1}}{{l}_{k}}}}\left( \frac{\sin \left( \gamma {{l}_{k}} \right) }{{{\eta }_{6}}R}-\left( {{\gamma }_{1}}^{2}\sin \left( \gamma {{l}_{k}} \right) +2\gamma {{\gamma }_{1}}\cos \left( \gamma {{l}_{k}} \right) -{{\gamma }^{2}}\sin \left( \gamma {{l}_{k}} \right) \right) \right) \\&{{a}_{61}}=-\alpha \sin \left( \alpha {{l}_{k}} \right) \left( E{{I}_{r}}\left( \frac{1}{{{\eta }_{1}}R}+{{\alpha }^{2}} \right) +\frac{G{{I}_{\theta }}}{R}\left( 1+\frac{1}{{{\eta }_{1}}R} \right) \right) \\&{{a}_{62}}=\alpha \cos \left( \alpha {{l}_{k}} \right) \left( E{{I}_{r}}\left( \frac{1}{{{\eta }_{2}}R}+{{\alpha }^{2}} \right) +\frac{G{{I}_{\theta }}}{R}\left( 1+\frac{1}{{{\eta }_{2}}R} \right) \right) \\&{{a}_{63}}={{k}_{1}}{{e}^{{{\beta }_{1}}{{l}_{k}}}}\cos \left( {{\beta }_{1}}{{l}_{k}} \right) +{{e}^{{{\beta }_{1}}{{l}_{k}}}}\left( E{{I}_{r}}{{a}_{631}}+\frac{G{{I}_{\theta }}}{R}{{a}_{632}} \right) \\&{{a}_{64}}={{k}_{1}}{{e}^{{{\beta }_{1}}{{l}_{k}}}}\sin \left( {{\beta }_{1}}{{l}_{k}} \right) +{{e}^{{{\beta }_{1}}{{l}_{k}}}}\left( E{{I}_{r}}{{a}_{641}}+\frac{G{{I}_{\theta }}}{R}{{a}_{642}} \right) \\&{{a}_{65}}={{k}_{1}}{{e}^{{{\gamma }_{1}}{{l}_{k}}}}\cos \left( {{\beta }_{1}}{{l}_{k}} \right) +{{e}^{{{\gamma }_{1}}{{l}_{k}}}}\left( E{{I}_{r}}{{a}_{651}}+\frac{G{{I}_{\theta }}}{R}{{a}_{652}} \right) \\ \end{aligned} \right. \end{aligned}$$

(A1)

$$\begin{aligned} \left\{ \begin{aligned}&{{a}_{66}}={{k}_{1}}{{e}^{{{\gamma }_{1}}{{l}_{k}}}}\sin \left( {{\beta }_{1}}{{l}_{k}} \right) +{{e}^{{{\gamma }_{1}}{{l}_{k}}}}\left( E{{I}_{r}}{{a}_{661}}+\frac{G{{I}_{\theta }}}{R}{{a}_{662}} \right) \\&{{a}_{631}}=\frac{{{\beta }_{1}}\cos \left( \beta {{l}_{k}} \right) -\beta \sin \left( \beta {{l}_{k}} \right) }{{{\eta }_{3}}R}-\left( \left( {{\beta }^{3}}-3\beta \beta _{1}^{2} \right) \sin \left( \beta {{l}_{k}} \right) +\left( \beta _{1}^{3}-3{{\beta }_{1}}{{\beta }^{2}} \right) \cos \left( \beta {{l}_{k}} \right) \right) \\&{{a}_{632}}={{\beta }_{1}}\cos \left( \beta {{l}_{k}} \right) -\beta \sin \left( \beta {{l}_{k}} \right) +\frac{{{\beta }_{1}}\cos \left( \beta {{l}_{k}} \right) -\beta \sin \left( \beta {{l}_{k}} \right) }{{{\eta }_{3}}R} \\&{{a}_{641}}=\frac{{{\beta }_{1}}\sin \left( \beta {{l}_{k}} \right) +\beta \cos \left( \beta {{l}_{k}} \right) }{{{\eta }_{4}}R}-\left( \left( 3\beta \beta _{1}^{2}-{{\beta }^{3}} \right) \cos \left( \beta {{l}_{k}} \right) +\left( \beta _{1}^{3}-3{{\beta }^{2}}{{\beta }_{1}} \right) \sin \left( \beta {{l}_{k}} \right) \right) \\&{{a}_{642}}={{\beta }_{1}}\sin \left( \beta {{l}_{k}} \right) +\beta \cos \left( \beta {{l}_{k}} \right) +\frac{{{\beta }_{1}}\sin \left( \beta \theta \right) {{l}_{k}}+\beta \cos \left( \beta {{l}_{k}} \right) }{{{\eta }_{4}}R} \\&{{a}_{651}}=\frac{{{\gamma }_{1}}\cos \left( \gamma {{l}_{k}} \right) -\beta \sin \left( \gamma {{l}_{k}} \right) }{{{\eta }_{5}}R}-\left( \left( {{\gamma }^{3}}-3\gamma \gamma _{1}^{2} \right) \sin \left( \gamma {{l}_{k}} \right) +\left( \gamma _{1}^{3}-3{{\gamma }_{1}}{{\gamma }^{2}} \right) \cos \left( \gamma {{l}_{k}} \right) \right) \\&{{a}_{652}}={{\gamma }_{1}}\cos \left( \gamma {{l}_{k}} \right) -\gamma \sin \left( \gamma {{l}_{k}} \right) +\frac{{{\gamma }_{1}}\cos \left( \gamma {{l}_{k}} \right) -\beta \sin \left( \gamma {{l}_{k}} \right) }{{{\eta }_{5}}R} \\&{{a}_{661}}=\frac{{{\gamma }_{1}}\sin \left( \gamma {{l}_{k}} \right) +\beta \cos \left( \gamma {{l}_{k}} \right) }{{{\eta }_{6}}R}-\left( \left( 3\gamma \gamma _{1}^{2}-{{\gamma }^{3}} \right) \cos \left( \gamma {{l}_{k}} \right) +\left( \gamma _{1}^{3}-3{{\gamma }^{2}}{{\gamma }_{1}} \right) \sin \left( \gamma {{l}_{k}} \right) \right) \\&{{a}_{662}}={{\gamma }_{1}}\sin \left( \gamma {{l}_{k}} \right) +\gamma \cos \left( \gamma {{l}_{k}} \right) +\frac{{{\gamma }_{1}}\sin \left( \gamma {{l}_{k}} \right) +\beta \cos \left( \gamma {{l}_{k}} \right) }{{{\eta }_{6}}R} \\ \end{aligned} \right. \end{aligned}$$

(A2)

$$\begin{aligned} \left\{ \begin{aligned}&{{b}_{42}}=\alpha \left( \frac{1}{R}+\frac{1}{{{\eta }_{2}}} \right) ,{{b}_{43}}={{\beta }_{1}}\left( \frac{1}{{{\eta }_{3}}}+\frac{1}{R} \right) ,{{b}_{44}}=\beta \left( \frac{1}{R}+\frac{1}{{{\eta }_{4}}} \right) \\&{{b}_{45}}={{\gamma }_{1}}\left( \frac{1}{R}+\frac{1}{{{\eta }_{5}}} \right) ,{{b}_{46}}=\gamma \left( \frac{1}{{{\eta }_{6}}}+\frac{1}{R} \right) {{A}_{6}}, {{b}_{51}}=\left( \frac{1}{{{\eta }_{1}}R}+{{\alpha }^{2}} \right) \\&{{b}_{53}}=-\left( {{\beta }_{1}}^{2}-{{\beta }^{2}}-\frac{1}{{{\eta }_{3}}R} \right) ,{{b}_{54}}=-2\beta {{\beta }_{1}},{{b}_{55}}=-\left( {{\gamma }_{1}}^{2}-{{\gamma }^{2}}-\frac{1}{{{\eta }_{5}}R} \right) \\&{{b}_{56}}=-2\gamma {{\gamma }_{1}}, {{b}_{62}}=\alpha \left( E{{I}_{r}}\left( \frac{1}{{{\eta }_{2}}R}+{{\alpha }^{2}} \right) +\frac{G{{I}_{\theta }}}{R}\left( 1+\frac{1}{{{\eta }_{2}}R} \right) \right) \\&{{b}_{63}}=\left( E{{I}_{r}}\left( \frac{{{\beta }_{1}}}{{{\eta }_{3}}R}-\left( \beta _{1}^{3}-3{{\beta }_{1}}{{\beta }^{2}} \right) \right) +\frac{G{{I}_{\theta }}}{R}\left( {{\beta }_{1}}+\frac{{{\beta }_{1}}}{{{\eta }_{3}}R} \right) \right) \\&{{b}_{64}}=\left( E{{I}_{r}}\left( \frac{\beta }{{{\eta }_{4}}R}-\left( 3\beta \beta _{1}^{2}-{{\beta }^{3}} \right) \right) +\frac{G{{I}_{\theta }}}{R}\left( \beta +\frac{\beta }{{{\eta }_{4}}R} \right) \right) \\&{{b}_{65}}=\left( E{{I}_{r}}\left( \frac{{{\gamma }_{1}}}{{{\eta }_{5}}R}-\left( \gamma _{1}^{3}-3{{\gamma }_{1}}{{\gamma }^{2}} \right) \right) +\frac{G{{I}_{\theta }}}{R}\left( {{\gamma }_{1}}+\frac{{{\gamma }_{1}}}{{{\eta }_{5}}R} \right) \right) \\&{{b}_{66}}=\left( E{{I}_{r}}\left( \frac{\beta }{{{\eta }_{6}}R}-\left( 3\gamma \gamma _{1}^{2}-{{\gamma }^{3}} \right) \right) +\frac{G{{I}_{\theta }}}{R}\left( \gamma +\frac{\beta }{{{\eta }_{6}}R} \right) \right) \\ \end{aligned} \right. \end{aligned}$$

(A3)